Mathlib v4.30 source proposition Nat.coprime_of_lt_prime

Identifier: F:ml430-nat-coprime-of-lt-prime-1978a919
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

The proposition declared as `Nat.coprime_of_lt_prime` in the pinned Mathlib v4.30 source.

Formal statement

∀ {n p : ℕ}, n ≠ 0 → n < p → Nat.Prime p → p.Coprime n

Dependencies

The graph shows direct ledger edges. Follow a node to open its artifact page.

6 direct dependencies 4 direct dependents

Evidence

kernel-Nat.coprime_of_lt_prime

Kind: kernel-term
Status: checked

Supports: ∀ {n p : ℕ}, n ≠ 0 → n < p → Nat.Prime p → p.Coprime n

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- coprime_of_lt_prime 2>/dev/null | grep -Ec '^Nat\.coprime_of_lt_prime[[:space:]]')" -ge 1

Evidence notes

`build_nat_prelude` admits `Nat.coprime_of_lt_prime` through the trusted `Kernel::add_declaration` gate, which re-checks the proof term against the stated type, so producing this row at all is a machine-checked proof. `nat_theorem_inventory` exits non-zero for a name that does not exist, and the `grep -c` count (tested `-ge 1`, not piped into `grep -q`) requires the admitted declaration to actually be printed. This declaration (`declare_coprime_of_lt_prime`, `nat_prelude/primes.rs`) already existed in the kernel before this lane started (landed in an earlier commit, `de2e39eee`), proving `∀ p a, prime_condition p → 0 < a → a < p → gcd a p = 1` directly from primality's divisor clause plus `gcd_dvd_right`/`gcd_dvd_left`/`le_of_dvd`/`lt_irrefl` -- a route that does NOT go through `coprime_or_dvd_of_prime` despite that being this fact's recorded `depends_on` edge (the dependency listing predates the direct proof and was not revisited). `Nat.Coprime p n` and `gcd n p = 1` are the same proposition up to argument order and Coprime's standard unfolding; this prelude spells coprimality directly as a `gcd = 1` equation, matching `coprime_of_bezout_one`'s convention noted in the declaration's own doc comment.

footprint-Nat.coprime_of_lt_prime

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the Nat prelude's trusted surface is empty

Checker command

cargo run -q -p axeyum-lean-kernel --example nat_axiom_inventory -- --require-axiom-free nat

Evidence notes

`nat_axiom_inventory --require-axiom-free nat` enumerates the built Nat environment and exits non-zero unless it admits no Axiom, Opaque or Quotient declaration (measured: axiom=0 opaque=0 quotient=0). A theorem cannot depend on a trusted declaration the environment does not contain, so an empty trusted surface bounds every individual theorem's footprint by [].

Provenance

{
  "date": "2026-08-18",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane parity-coprime",
  "source": "statement-only extraction of `Nat.coprime_of_lt_prime` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.coprime_of_lt_prime`",
      "where": "mathlib4 commit c5ea00351c28e24afc9f0f84379aa41082b1188f (v4.30.0)",
      "year": 2026,
      "attribution": "the proposition was read from the pinned statement-only inventory; the proof term and tactic trace were not consulted"
    }
  ]
}